강사:

Olivier L. Burdet

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Hello, in this video, I will talk about the beams with cantilevers, and about Gerber beams. So far, in these videos, we have mostly looked at what happens in Gerber trusses. And we now want to look at what happens in Gerber beams. We will see what is the effect of the cantilevers, how the combination of the internal forces works in beams with cantilevers, how to build Gerber beams, in particular the Gerber joints between the elements, then we will see how it can be interesting to use Gerber beams in agreement with the construction process. Here, we are going to compare various configurations of beams. We will always have an arch and a cable. This arch and this cable form if we have a simple beam here, subjected to a uniformly distributed load over the whole length of the beam, a uniform load q. And then you can see that here the beams have a greater length, and in all the cases, that is going to be distributed over the whole length of the beam, not only on the central part, but here, since we have not a beam which goes further, here we have a length L. And then, we are going to compare each time the length of the cantilever l' to the reference length L. So, initially, we have a simple beam, without any cantilevers, l' over L equals to zero. We have a rise which is equal to f, which is the maximal distance between the arch and the cable, which is obviously at mid-span. And then we have a radius of curvature... an upward curvature, with a radius of curvature which indicates it. If we create two small cantilevers of 0.2 times the span, on the left and on the right, then the arch takes this shape, again under the uniform loads distributed over the whole length of the beam, and then the cable is here. Between the arch and the cable, we have a rise which decreased to 0.84f at mid-span, therefore the internal forces are smaller than before, however, we have a small rise of 0.16f, so internal forces, while before, in theory, the internal forces were zero at the supports since the distance was zero. For the curvature, we still have an upward curvature, over the whole length of the beam, even if we can see a little something which tends to bend in the other direction at the ends. If we lengthen the cantilever to approximately 0.35L, what we can notice is that the cable is located at mid-depth between the upper part and the lower part of the arch. We can see it here, with 0.5f upwards, and 0.5f downwards at the supports. You can note that now, the internal force is half of what it was before. That is a quite interesting configuration to have a beam with cantilevers which are about 0.35, approximately 1/3 of the span. If we increase the length of these cantilever to 0.4 times the span, then the configuration has this shape, that is to say that the arch tends to be lower than the cable, except in the central part. That is to say that we have 0.33f, 1/3 of the initial rise, at mid-span, and 2/3 of the initial rise on the supports. The curvature - I have forgotten to do it for the case of 0.35 - here we had a curvature upwards in the central part, and here a curvature downwards, on the sides. For the case l'/L = 0.4, we still can see a bit of upward curvature in the central part, but we especially have a downward curvature in most of the structure, from here to here. Finally, the last step here is to have l'/L = 0.5, that is to say cantilevers which are as long as 0.5 times the span between both supports. And in this case, the arch is completely under the cable, As a result, as we can see it here, the rise is here integral. The rise is equal to f, but in this case, it is entirely on the supports, at mid-span, the internal forces are equal to zero. If we look at the curvature, over the entire length of the beam, we now have a downward curvature. The upward curvature has entirely disappeared, there is only a downward curvature now. So we can see that the effect of the cantilevers on a beam is very significant, it can significantly reduce the internal forces and change the deformations. The cantilevers of about 1/3 of the span enable to decrease the internal forces as much as possible, dividing them by two. What we have just seen about beams with cantilevers, the animals know it, at least their skeleton is built in a way which demonstrates that it is efficient under the effect of these loads. An animal is schematized here as a beam with two cantilevers, and we can see here, the muscles are tied at the top of its spine, we have system of muscles which is like this: the rear is very limited here with a very small tail, but we can see that its system is well adapated. In the first case, it is a cow which is characterized by a very heavy head, it is thus necessary to have a large depth here, since that is here that the internal forces are maximal to support the weight of the head. The animal on the bottom, I think you have guessed what it is, that is a diplodocus. Here we have a different configuration with a very long tail, then again a spinal column able to carry compression, and then a relatively small head. So the internal forces are going to... the muscles are going to be tied at the top of the crest, here. And we can notice that there is a good similarity between the shape which is given to this animal, and the internal forces which are likely to act on it: essentially uniformly distributed loads, probably slightly variable, there are a bit more load in the middle than at the ends, but there are no large loads like the one of the head of the cow. In this video, I show you how we can build a Gerber beam. We have already seen this for trusses with the example of the Forth bridge. We can then build the left beam and the right beam with cantilevers - for example, above a highway - then afterwards, we can place the central part which misses and which will be able to carry loads and to distribute them onto the elements on the left and on the right. So we can notice that the middle beam takes support on the end of the cantilevers of the left beam, and of the right beam. Obviously, we cannot drive on such a structure, so we must create a joint to make what we have drawn here: that is to say, we have drawn something with a support which is placed on the top, but obviously, a vehicle or a pedestrian could not pass here. So what we want to do is to create a joint which enables traffic to pass without problem, while corresponding to what has been decided. So we are going to create the first beam which is the one on which the second beam is going to take support, then we are going to put a support here. That could be for example a neoprene block, we have seen that during the first semester. Then, we will have the second part of the second beam which will have a complementary shape to take support on this element. I will show you soon a picture of such a detail. So here, that is a support. And obviously, with this, we have created a flat surface on which we can pass, and something which also respects the fact that the arch and the cable must cross in one point, that is going to be this point here, which is the support. There is a problem with this configuration when we apply it to bridges. Why? When we have a bridge, we are going to have here a water inlet, H2O, plus, if we are in a northern country for example, salt, Na Cl, or chlorides in general. With the possibility here to go through the joint, the water is going to attack these zones here. These zones are not easy to inspect or to repair. As a consequence, the bridge is going to deteriorate in this zone here. In this case, we are going to avoid Gerber joints. But these joints are absolutely practicable in other types of constructions which are not exposed, or less exposed to the aggressive action of water and salt: for example, in buildings, it is a practicable solution. So here, without too much detail because we have just seen it, we are going to create this Gerber joint, on the left and on the right, to enable the support of the beam such as we have designed it. - I draw it in one go not to waste time - So, we have what enables us to make what I have demonstrated to you with the model, with a beam which is now a Gerber beam. Let's just check if it is statically determinate or not. So here we have two support reactions. Here we have one, here one, and here too. So we have a total of five support reactions. Let's now count the number of bars: we have one, two, three bars. Let's now count the number of nodes: we have one node here, - but not here because that is not the end of a bar - a second node here, a third node here, a fourth one. I thus have two times four which is equal to eigth, and therefore the structure is statically determinate. So Gerber beams, because we have added a certain number of joints, remain statically determinate while normally, if we have a beam which has more than two supports, it would be statically indeterminate. Here I have an example of Gerber slab, which I have observed at the Bern train station. Here we have a Gerber joint, another Gerber joint on the other side. It is quite clear that these elements on the left and on the right have been built first and that this slab has been added later. Here, there is no joint problem since this is a zone which is not exposed to salt, it is even quite dry. So that is a good solution which enables a very rational construction. You remember that in the first example of beam with cantilever, I could, increasing the load at the end of the cantilever, lift up the opposite supports. When I have a Gerber beam, I can also have this configuration in which the end supports lift up. - of course, it is necessary to apply a larger force - That is not necessarily a problem, but in some configurations, it is necessary to be careful with the construction process to avoid it happening. If it had to happen, that could be catastrophic. The solution is often to ballast, or to retain the supports to prevent them from lifting up. Here we have an example from the 1950's of a bridge in Italy with Gerber joints. We cannot see them well, so I am going to evidence them. Here we have a Gerber joint, here too, and then here too for each of these beams. So we have two families of beams: we have already seen this configuration when we were looking at Gerber trusses. We have these large beams here, with cantilevers, and Gerber joints at the end. So it corresponds here to this configuration here on the top. And then between these beams, we have beams which have been placed at a later stage. - I draw them in blue - This kind of construction was very popular in Italy because it really makes sense to use Gerber joints since it enables to build very quickly. Obviously, the previously notified problem for these Gerber joints remains valid: that is something which we did not know at that time, but that now we would avoid building. So here, we have parts of Gerber beams. In this applet, I have already inserted all the loads, as well as the initial values for the support reactions. I am going to show you how we can obtain the solution of this structure. So I call the funicular polygon, then I select, using the Control button, the support reactions, which are obviously not correct. What will be necessary is that the arch and the cable cross in all these points, so, for this, I am going to, pressing the Shift button , slowly increase the support reaction, then we can see that as we go along, it goes down, and we did not exactly reach it yet, still a little step, one try. That is not easy, we have to be accurate. Here, that is not that bad. We can see that here the arch and the cable cross, here that is quite close, here too, and so on. So with this, we have just solved a system where there was four unknown forces, four support reactions, in such a way that the system works as expected, like a Gerber beam. You will have the opportunity to do the same kind of work in your exercises. In this video, I show you how to build a Gerber beam with a different system: usually, we built end beams, then we placed an element in the middle. Here, we start by the left, we create two supports, then a beam with a cantilever, an additional support on which we place the following beam, which also has a cantilever on which the last beam is going to lean, carried as well by the last support. That corresponds in a quite accurate way to the construction sequence of a bridge. For example, we can imagine to have here a first beam which we have built. Under this beam, we have a scaffolding, with temporary supports, something like this. Then we are going to be able to take off this scaffolding, to move to the second beam which we are going to build here. So we will have disassembled the scaffolding, and we will be able to re-assemble it under the second beam, here, with again some posts. Thanks to this way to proceed, we will succeed to build the entire structure, for example a bridge, but it can be something else, in a logical way, using the parts which are already built as supports for the parts which still need to be built. Here on the bottom, I have drawn, as seen previously, a practical solution for this configuration of Gerber beams, with intermediate supports. Again, that is not a solution which is used nowadays for bridges, but for other types of structures, that is not a problem. Finally, here, a configuration with slightly particular Gerber joints for the Confederation bridge in Canada. What we can see under is the estuary of the Saint-Lawrence River, which can be seriously frozen during part of winter. On the other hand, that is a zone by which big ships, oil tankers, etc, pass. What could happen, in one of the scenarios which have been considered, is that one of these ships hit the bridge which would risk to be damaged. What is the strategy which has been set up for the bridge not to be entirely lost in the event of an impact? What we have done is that we have created Gerber joints at the ends of segments which are delimited every two piers. Between these Gerber seals, we have placed a structural element, that element here, in the middle, so every two elements, there is a blue element, and so forth, over all the length of the bridge. If an impact had to occur, and that, let's say, we lost this part, in orange, because there had been an impact on one of these two piers, this orange part would thus fall down, and also the two blue parts since they take support on the orange part, but the rest of the bridge would remain stable, because the other elements of the bridge would not be affected: this Gerber element has been placed with a crane later on, so we could let it fall down into the water and rebuild the orange element to make the bridge continuous again. That is an unusual strategy, as I said, usually we do not really like Gerber joints in bridges, but in this case here, there is a logic to respect. In this video, I have shown you the effect of cantilevers on beams, how the combination of beams with cantilevers, of intermediate beams, is very interesting, how, in Gerber beams, we solve the problem of details using Gerber joints, and some examples of application.